The Research On Well-posedness Of Solutions For Two Classes Of Nonlinear Parabolic Equation

This paper investigates the well-posedness of solutions for a class of nonlinear parabolic equations at high initial energy level and a class of nonlinear parabolic equation with strong damping term on the cone manifold to reveal the influence of the framework of functionals and the initial data on the qualitative behavior of solutions for nonlinear parabolic equation by applying the comparison principle and potential well method.Chapter 2 shows the well-posedness of solutions for a class of nonlinear parabolic equa-tion at high initial energy level. There are a lot of results on the well-posedness of solutions for this problem considered in this chapter at the low initial energy level and critical initial energy level, however, we can not obtain the global existence and finite tim blow up of solu-tions at high initial energy level by using the potential well method because of the lack of the invariant sets at high initial energy level. In this chapter, we overcome this difficulty by us-ing the potential well method, the semigroup of operators and the comparison principle. This chapter analyzes the characteristics of the solution of steady state equation corresponding to the evolution system by constructing the potential well structure of the considered problem,which is used to construct the semigroup operator theory comparison principle and by using the strong maximum principle for the homogeneous parabolic equation, we obtain the initial data such that the solution exists globally or blows up in finite time at high initial energy level.Chapter 3 investigates the global existence and finite time blow up of solutions for the initial boundary value problem to a class of parabolic equations with strongly damping on cone degenrate manifold. Cone manifold is a kind of topological space which has the ef-fect of boundary degradation. This chapter constructs the variational structure with respect to the nonlinear damped parabolic equations on cone degenerate manifold and divides the re-lationship between the well-posedness of solutions and the energy into three cases, and then generalizes the potential well method to the nonlinear damped parabolic equations on cone space. This chapter starts from the basic characteristics of cone space to prove the global ex-istence of solution with low-initial energy by using the Galerkin method and the boundedness of approximate Galerkin solution. By introducing a new auxiliary function and employing the classical concave method we obtain the finite-time blow-up result of solutions. Later we use the multiplier method to show the sets on the initial data such that the global solutions decay exponentially. In the end by employing the scale idea we deal with the classical case in the low-initial case.